Respuesta :
Given:
Degree of polynomial = 3
Zeros are -3,-1, and 4.
The leading coefficient is -4.
To find:
The polynomial.
Solution:
The general form of a polynomial is
[tex]P(x)=a(x-c_1)^{m_1}(x-c_2)^{m_2}...(x-c_n)^{m_n}[/tex]
where, a is a constant, [tex]c_1,c_2,...,c_n[/tex] are zeros with multiplicity [tex]m_1,m_2,...,m_n[/tex] respectively.
Zeros of the polynomial are -3,-1, and 4. So,
[tex]P(x)=a(x-(-3))(x-(-1))(x-4)[/tex]
[tex]P(x)=a(x+3)(x+1)(x-4)[/tex]
[tex]P(x)=a(x^2+3x+x+3)(x-4)[/tex]
[tex]P(x)=a(x^2+4x+3)(x-4)[/tex]
[tex]P(x)=a(x^3+4x^2+3x-4x^2-16x-12)[/tex]
[tex]P(x)=a(x^3-13x-12)[/tex]
[tex]P(x)=ax^3-13ax-12a[/tex]
Here, leading coefficient is a.
The leading coefficient is -4. So, a=-4.
[tex]P(x)=(-4)x^3-13(-4)x-12(-4)[/tex]
[tex]P(x)=-4x^3+52x+48[/tex]
Therefore, the required polynomial is [tex]P(x)=-4x^3+52x+48[/tex].
